This paper examines the possibility that the elimination of synapses from cells arises from a competition between the nerve terminals for trophic molecules made available by the cells. This idea is applied to the elimination of synapses that occurs during the polyneuronal innervation of muscle cells which accompanies both the development and reinnervation of muscles. In the proposed model, each motorneuron makes the same amount of receptor in its soma for a trophic molecule provided in limited quantities by each muscle cell; this receptor is then distributed to the collateral terminals of the motorneuron in concentrations proportional to the amount of receptor made in the soma by the motorneuron; the more collateral terminals initially possessed by a motorneuron the less will be their concentration of receptor. The receptors in the several collateral terminals on a muscle cell then complete for the trophic molecule provided by the muscle, and terminal growth is proportional to the number of receptor--trophic-molecule bonds formed. An autocatalytic effect has been introduced whereby the increase in size of a terminal accelerates the rate by which the trophic molecule is made available to that terminal for bonding with its receptors. In addition, the affinity between nerve terminal receptors and muscle molecules can be varied in the model. Finally, motorneuron cell death has been analysed as the elimination of neurons that have insufficient terminal area to take up a growth factor in amounts that will allow for the survival of the neuron.